Understanding Annihilator Method for y'' + 4y = 0

[s3a]

Homework Statement

(Snapshot time = 1:02)

Homework Equations

Annihilator method.

The Attempt at a Solution

In the snapshot at time = 1:02 of this video, I don't get the y'' + 4y = 0 despite the annotations the author placed. Also, a comment in the video says "you skipped finding and applying the actual annihilator." Could someone please elaborate on all this since I'm having major trouble with this topic?

Any help in understanding this would be greatly appreciated!
Thanks in advance!

 

[HallsofIvy]

In general, you solve a homogeneous ODE with constant coefficients by solving the "characteristic equation". For a second order equation, with real coefficients, that will be a quadratic equation with roots of the form a+ bi and a- bi. The corresponding solutions to the ODE will be $e^{(a+ bi)x}= e^axe^{bix}$ and $e^{(a- bi)x}= e^xe^{bix}$. As you have probably learned, $e^{bix}= cos(bx)+ i sin(bx)$ (you can get that by comparing the Tayor's series of $e^{bix}$ with the Taylor's series of sin(bx) and cos(bx)). In particular, sin(2x) and cos(2x) correspond to $e^{2ix}$ and $e^{-2ix}$ which means that the "characteristic solutions" are 2i and -2i which, in turn, come from the characterstic equation $r^2= -4$ or $r^2+ 4= 0$ which corresponds to the ODE $y''+ 4y= 0$. The point is that, to use the "Annihilator method" (also called the "method of undetermined coefficients"), the function of x, here, sin(2x), must be a possible solution of a homogeneous differential equation with constant coefficients and that is true here- sin(2x) is a solution to y''+ 4y= 0.

As for "you skipped finding and applying the actual annihilator", I watched this entire video and did not find those words.

 

[Bohrok]

The OP said it was a comment that said "you skipped finding and applying the actual annihilator."
I think s/he wanted elaboration on using annihilators for the problem.